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Old 2020-08-27, 13:09   #37
Dec 2011

11×13 Posts

Originally Posted by Dr Sardonicus View Post
There is also the formula in this "Chinese paper" on Aurifeuillian factorizations of numbers of the form

M^{n} \; \pm \; 1
First, I admit I am having a bit of difficulty with the notation in the 1999 "Chinese paper". What I am wondering is whether or not this paper really describes a "new class" of Aurifeuillian Factorizations.

Jumping to the example at the end, the paper demonstrates the factorization of a certain 362-digit number into the product of a 181-digit number times a 182-digit number. Although I was unaware of this 1999 paper, my software cracked the Phi function (Brent's notation) of 44^253+1 into the same 181- and 182-digit L and M components using the methods of Brent's 1993 and 1995 papers, which I referenced a few posts back.

The M component had a 46-digit factor, which I obtained via ECM with B1=43M on January 16, 2013. The remaining 116-digit composite was factored by me using GNFS on January 21, 2013. (The penultimate factor of the L component contained 21 digits.) [I'm not suggesting I was the first to perform any of this factorization -- merely that I had an interest and I happen to have factored this number.]

Based on one example, the results of this "new class" are the same as results using traditional methods. Does anybody know if there are any cases where this method can find additional factorizations not found by a good understanding of Brent's methods? If so, I'll try to more fully understand the referenced 1999 paper.

Last fiddled with by rcv on 2020-08-27 at 13:22
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